Effect of small interpulsar distances in stochastic gravitational wave background searches with pulsar timing arrays

Abstract

One of the primary objectives for pulsar timing arrays (PTAs) is to detect a stochastic background generated by the incoherent superposition of gravitational waves (GWs), in particular from the cosmic population of supermassive black hole binaries. Current stochastic background searches assume that pulsars in a PTA are separated from each other and the Earth by many GW wavelengths. As more millisecond pulsars are discovered and added to PTAs, some may be separated by only a few radiation wavelengths or less, resulting in correlated GW phase changes between close pulsars in the array. Here we investigate how PTA overlap reduction functions (ORFs), up to quadrupole order, are affected by these additional correlated phase changes, and how they are in turn affected by relaxing the assumption that all pulsars are equidistant from the solar system barycenter. We find that in the low-frequency GW background limit of f ∼ 10 −9 Hz, and for pulsars at varying distances from the Earth, these additional correlations only affect the ORFs by a few percent for pulsar pairs at large angular separations, as expected. However, when nearby (order 100 pc) pulsars are separated by less than a few degrees, the correlated phase changes can introduce variations of a few tens of percent in the magnitude of the isotropic ORF, and much larger fractional differences in the anisotropic ORFs-up to 188 in the m ¼ 0, l ¼ 2 ORF for equidistant pulsars separated by 3°. In fact, the magnitude of most of the anisotropic ORFs is largest at small, but nonzero, pulsar separations. Finally, we write down a small angle approximation for the correlated phase changes which can easily be implemented in search pipelines, and for completeness, examine the behavior of the ORFs for pulsars which lie at a radiation wavelength from the Earth.

The local nanohertz gravitational-wave landscape from supermassive black hole binaries

“…Since upper limit curves, and not detection curves, were reported by Ref (Babak et al 2016), we use the upper limit as a proxy for detection to underline the importance of the pulsar and SMBHB sky location. Indeed, the PTA response to CGWs is maximal when the GW source lies very close to the pulsar (Detweiler 1979;Mingarelli & Sidery 2014).…”

Limits on Anisotropy in the Nanohertz Stochastic Gravitational Wave Background

“…This frequency shift is integrated over time to give the induced timing residuals, rðtÞ ≡ R t δνðt 0 Þ=ν 0 dt 0 , which are cross-correlated between pulsars in an effort to boost the detection probability of GW signals at Earth. The expectation value of the cross-correlated timing residuals between pulsars a and b is proportional to the overlap reduction function (ORF, Γ ab )-a dimensionless function that quantifies the response of a pair of pulsars to the stochastic GW background [26,27]. In this Letter, we use analytically computed anisotropic ORFs [28,29] and recently developed Bayesian techniques [30] to constrain the angular power distribution of the SGWB.…”

Phase-coherent mapping of gravitational-wave backgrounds using ground-based laser interferometers

“…The above expression for R A (f,k) is valid in the small-antenna limit, which is appropriate for groundbased interferometers like LIGO, Virgo, etc. (see e.g., [20] for this discussion in the pulsar timing context). The length of the interferometer arms do not enter the expressions for the response functions in this limit.…”

“…Since the coordinate system for the response functions in (18) was not chosen in any particular orientation relative to the unit vectorsû andv, Eqs. (20) are valid in an arbitrary translated and rotated coordinate system, provided we use the angles forû,v, andx 0 as calculated in the rotated frame.…”

“…Eqs. (15) and (20), where x 0 is the location of the detector at the time t at which the measurement is made. For cross-correlation measurements between a pair of detectors, the correlated response depends on the location of the detectors x 1 and x 2 only via their difference ∆ x ≡ x 2 − x 1 , which is independent of choice of origin.…”